Decidability and computability of certain torsion-free abelian groups. (English) Zbl 1211.03063

Downey, Rodney G.; Goncharov, Sergei S.; Kach, Asher M.; Knight, Julia F.; Kudinov, Oleg V.; Melnikov, Alexander G.; Turetsky, Daniel

Notre Dame J. Formal Logic 51, No. 1, 85-96 (2010).

The paper investigates the computable properties of torsion-free abelian groups of the form \({\mathcal G}_S = \bigoplus_{n \in S} {\mathbb{Q}}_{p_n}\), for sets \(S \subseteq \omega\), where \(p_n\) is the \(n\)-th prime number and \({\mathbb{Q}}_p\) is the subgroup of \(({\mathbb{Q}}, +)\) generated by the numbers \(1/p^k\), for \(k \in \omega\). It is shown that \({\mathcal G}_S\) has a decidable copy if and only if \(S\) is \(\Sigma_2^0\) and has a computable copy if and only if \(S\) is \(\Sigma_3^0\).

Reviewer: Marius Zimand (Towson)

MSC:

03D45	Theory of numerations, effectively presented structures
03B25	Decidability of theories and sets of sentences
20K15	Torsion-free groups, finite rank
20K20	Torsion-free groups, infinite rank

Keywords:

completely decomposable torsion-free abelian groups; coding in groups